INTEGRATION OF OPTIMAL CLEANING SCHEDULING AND …heatexchanger-fouling.com/papers/papers2019/29... · corrects for dynamic process changes, plant disturbances and model mismatch

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INTEGRATION OF OPTIMAL CLEANING SCHEDULING AND CONTROL FOR

FOULING MITIGATION IN HEAT EXCHANGER NETWORKS:

A RECEDING HORIZON APPROACH

*F. Lozano-Santamaria1, and S. Macchietto1 1 Department of Chemical Engineering, Imperial College London, South Kensington Campus, London, SW7

2AZ, UK ([email protected])

ABSTRACT

Fouling mitigation is paramount to maintain a

reliable, efficient, and economic operation of heat

exchanger networks in the food, water, chemicals

and refining industries. Two common mitigation

strategies are controlling the flow distribution in the

network, and optimizing the periodic cleaning of

units. Due to high complexity, these are currently

addressed separately (sequentially). This paper

presents for the first time an online solution that i)

exploits the strong synergy between the two

strategies by optimizing them simultaneously and ii)

corrects for dynamic process changes, plant

disturbances and model mismatch by using a

nonlinear model predictive control scheme.

A network-wide estimation model exploits

measured and soft-sensed past data to generate an

accurate estimation of the current state and reliable

performance predictions over a long range. The

prediction model is then solved to optimize both

control and cleaning schedules. Estimation and

prediction models are continuously updated in a

receding horizon scheme. The significant economic

benefits achieved with this online solution are

demonstrated via a case study that captures all the

relevant elements of a refinery preheat train.

INTRODUCTION

Fouling in refining applications has been a

problem of major concern through the years. It has a

big impact in the operation, safety, and economics

of the process. Fouling is especially important in the

preheat train, where all the crude processed in a

refinery is heated up from atmospheric conditions to

the high temperature (~ 360°C) required in the crude

distillation unit (CDU). At these conditions crude oil

has a high fouling propensity, and a small decay in

the efficiency of the heat exchangers incurs large

costs. Eliminating or reducing fouling has economic

benefits for the operation as it leads to significant

energy savings. Fouling mitigation strategies

include design options (the use of antifouling agents,

surface coatings, heat exchanger inserts, network

retrofits), and operational options, discussed below

[1], [2]. So far, no single design approach has proved

capable of eliminating fouling completely in the

preheat train of a refinery, so that operational

mitigation alternatives still play an important role.

The most common operational options are control of

the flow distribution in the network (bypasses, flow

splits in parallel branches, e.g. Fig 1), and the

periodic cleaning of units. Flow control can reduce

fouling rates or partially remove the deposit by

increasing the sheer rate in certain units. Cleaning a

unit (offline or online) removes the deposit and

restores its thermal and hydraulic performance, but

takes it out of service for a period. Both generate

complex and hard to predict thermal and hydraulic

interactions in the network.

HEX1

Crude

HEX2A

HEX2B

HEX2C

Sp_crude Mx_crude

Kero

LGO

VR

Furnace

Fig 1. Small refinery heat exchanger network.

Several studies in the literature have looked at

the optimal cleaning scheduling problem on its own

(e.g. [3], [4]), while some analyzed the flow

distribution problem alone (e.g. [5], [6]). Only more

recently the integration of both at the same decision

level has been tackled [7], highlighting their strong

synergy and large benefits arising. The above

approaches differ in the way of modelling the heat

exchangers (e.g. lumped parameter models, cell

models) [8], in the fouling model used (e.g. semi-

empirical, empirical) [9], in the solution strategy

(e.g. deterministic optimization, stochastic

optimization) [10], or in the problem simplifications

adopted (e.g. linear approximation, cyclic operation)

[11], [12]. However, all cases assume that: i) the

Heat Exchanger Fouling and Cleaning – 2019

ISBN: 978-0-9984188-1-0; Published online www.heatexchanger-fouling.com

mailto:[email protected]

2

operating conditions in the process remain constant

or are known a-priori over long-time horizons (> 1

year), and ii) the model can predict the operation

perfectly for such horizon. In actual refinery

operations, the first condition does not hold, as

process conditions change frequently (for instance,

crude blends change every few days). The second

condition is very unlikely to hold for such long-

horizons in the presence of disturbances, and no

model is ever perfect. For these reasons, an offline

fouling mitigation solution developed at any one

time will not capture future variability in the

process, will soon get out of date and will lead to

suboptimal, possible even unfeasible operation.

Considering process variability in the decision

making would result in better fouling mitigation

actions. Acknowledging that any prediction model

has limitation and its prediction capabilities may

extend until certain point, one approach is to update

models periodically.

In this paper a new scheme and solution for

online fouling mitigation are presented, following

the principles of model predictive control (MPC)

used in control engineering. This approach uses past

primary measurements of the plant to regularly

update a prediction model of the plant performance,

and that model is used in the decision-making

process. Here, the prediction model is used to

optimally define an operation plan comprising a

schedule of cleaning actions (which exchanger to

clean and when) and time profiles for the flow split

distribution. The operation plan is implemented, and

then periodically updated online in a closed loop.

OPTIMIZATION COMPONENTS

This section briefly describes the key concepts

of model based receding horizon control. A

formulation of three optimization problems for

fouling mitigation is then introduced. The

performance of the on-line implementation depends

strongly on the update of these models from plant

data, their efficient solution, and implementation of

their predictions.

The on-line fouling mitigation methodology

proposed is based on two key concepts from process

control: 1) a nonlinear predictive controller (NMPC)

that predicts the future behavior of the plant and

defines the actions to take (manipulated variables)

[15], [16], and 2) a moving horizon estimator

(MHE) that updates the model parameters using past

data to ensure a good quality of prediction [17]. Fig

2 shows how these two elements work and interact.

Here we use an economic oriented NMPC instead of

the classic NMPC with a tracking objective

function. At each sampling time 𝑡∗ a solution of the

MHE problem defines the model parameters and

gives an estimation of the actual conditions of the

process at time 𝑡∗. These initial conditions are not

the current measured values from the plant because

they have noise or are not even measured (e.g. the

deposit thickness). The MHE problem is defined

over a past estimation horizon, PEH, that contains a

number N of measurement points. Next, the NMPC

problem is solved, the solution of which is the

optimal future actions (𝑢) over a future prediction

horizon (FPH), and corresponding prediction of the

future states (𝑥) of the plant. Although the

predictions are available for a long horizon, only the

first control action is implemented, and the rest are

discarded. At the next sampling time (𝑡 + Δ𝑡), with

Δ𝑡 much shorter than the prediction horizon, the

process is repeated. New measurements are

available to update the model by solving the MHE

problem, and new control actions are determined

and implemented by solving the NMPC problem.

Note that both the MHE and the NMPC operate in a

receding horizon scheme.

Fig 2. NMPC and MHE schematic description. a)

Past measurements and predictions at 𝑡 = 𝑡∗, b)

Past measurements and predictions at 𝑡 = 𝑡∗ + 𝛥𝑡

These two elements are the key ideas used in the

online approach for fouling mitigation. While the

MHE estimates the fouling model parameters and

current fouling state, the NMPC defines the actions

to take in order to minimize the effects of fouling

and maximize energy recovery. The fouling

mitigation problem is divided in two layers based on

the time scale of the process: 1) a scheduling layer,

and 2) a control layer. The scheduling layer captures

the operation over long time (~months). The

decisions involved in this layer have a low

frequency (which units to clean and when). On the

other hand, the control layer capture more rapid

variation of the process (~daily) and decisions in this

Time𝑡 = 𝑡∗

𝑃𝐸𝐻

𝑀𝐻𝐸 𝑁𝑀𝑃𝐶

min (𝑥𝑖 − 𝑥𝑖 )2

𝑖∈𝑁

min 𝐽(𝑥,𝑢) 𝑥

𝑢

𝑥𝑜

} Past measurementsEstimation of initial condtions

Predicted manipulated variables

Only control action

implemented

Time

𝑃𝐸𝐻

𝑀𝐻𝐸 𝑁𝑀𝑃𝐶

min (𝑥𝑖 − 𝑥𝑖 )2

𝑖∈𝑁

min 𝐽(𝑥,𝑢) 𝑥

𝑢

𝑡 = 𝑡∗ + Δ𝑡

a)

b)

𝐹𝑃𝐻

𝐹𝑃𝐻



3

layer are taken at high frequency (setpoints to flow

control valves). For each of these layers an

optimization problem that aims to minimize the total

operation cost over a future horizon is formulated.

Although models in the two layers share the same

objective and are similar in structure, the decision

variables, length of the prediction time, and

mathematical complexity of each are different.

The main building blocks for an accurate model

of heat exchanger networks under fouling are:

• A heat exchanger model to predict its

performance (heat duty, outlet streams

temperature, and pressure drop).

• A fouling model to predict the deposit time

evolution and how it is affected by the

operational conditions.

• A network model that defines dynamically

the connectivity between different units

(including stream mixers and splitters).

• An objective function that quantifies the

performance of the whole network to

evaluate the effectiveness of fouling

mitigation alternatives.

Optimization problems at the scheduling layer

and control layer share these building blocks. A

lumped parameter model, the P-NTU model, is used

to describe the operation of each heat exchanger.

This model calculates the heat duty and streams

temperature as a function of the inlet conditions and

design specifications of the unit (e.g. number of

passes, number of tubes) [13]. The semi-empirical

Ebert-Panchal model [14] is used to model the crude

oil fouling in each unit. This model predicts the

fouling thermal resistance as a function of the

operating conditions of the unit, and its parameters

can be tuned to match plant observations. The

hydraulic effects of fouling (pressure drop) are

calculated from the deposit thickness, taking into

account the curvature effects of the surface in the

deposit formed. The heat exchanger network is

modelled as a multigraph in which each stream type

(e.g. crude oil, LGO) defines a graph that connects

the nodes, and two graphs interact only at the heat

exchanger nodes. This formulation allows defining

the mass flow rate through the network, and setting

split ratios and inlet flow rates of different streams,

as detailed in [7].

The final element of the optimization model is

the objective function (OF), which in this case is the

minimization of the total operating cost due to

fouling over the time horizon (Eq. 1). The first term

in OF represents the energy cost, which is

proportional to the furnace duty. Fouling in the

exchangers requires more fuel to be used in the

furnace to maintain a desired inlet temperature to the

crude distillation unit. The second term accounts for

the cost of carbon emissions, assumed proportional

to the furnace duty. The final term is the cost

associated with all the cleanings performed on all

units.

𝐽𝑜𝑝 = ∫ [𝑃𝑄𝑄𝑓 + 𝑃𝐶𝑂2𝑚𝐶𝑂2𝑄𝑓]𝑑𝑡𝑡𝑓

𝑡0

+ 𝑃𝑐𝑙,𝑡,𝑖𝑦𝑡,𝑖𝑖∈𝐻𝐸𝑋𝑡∈𝑇𝑖𝑚𝑒

(1)

The optimization problem solved in the NMPC

at the scheduling layer is defined by Eq. 2. The

decision variables are the starting time of cleanings

(𝑡𝑠,𝑝,𝑖∀𝑝 ∈ 𝑃𝑒𝑟𝑖𝑜𝑑𝑠, 𝑖 ∈ 𝐻𝐸𝑋), the assignments of

cleanings to units (𝑦𝑡,𝑖∀𝑡 ∈ 𝑇𝑖𝑚𝑒, 𝑖 ∈ 𝐻𝐸𝑋), and

the flow rates in the network (𝑚𝑡,𝑎∀𝑡 ∈ 𝑇𝑖𝑚𝑒, 𝑎 ∈

𝐴𝑟𝑐𝑠). Flow rate decisions are considered together

with the cleaning scheduling variables because of

their strong interaction [18]. Although flow control

is implemented at a different level, flow rates

influence scheduling decisions in the upper layer,

and cannot be omitted. Constraints include: the

network connectivity equations (mass and energy

balance at each node), the heat exchanger model and

the fouling model for all units, operational

constraints (e.g. firing limit of the furnace), a

continuous time representation of time, any

scheduling constraints (e.g. “clean certain

exchangers simultaneously”, “do not clean a specific

unit in a specific period”), and the formulation of

disjunctions. The latter are used to model logical OR

decisions, for example that an exchanger can be

either in a “cleaning” or “operating” state at a

specific time. This is an MINLP problem with a

large number of binary decision variables and

nonlinear terms, over long prediction horizons (0.5

– 4 years). The reader is referred to [7] for more

information about the continuous time formulation

and the solution strategy.

min𝑡𝑠,𝑦,𝑚

𝐽𝑆 = 𝐸𝑞. 3

𝑠. 𝑡. 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝐻𝑒𝑎𝑡 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒𝑟𝑠 𝑚𝑜𝑑𝑒𝑙 𝐹𝑜𝑢𝑙𝑖𝑛𝑔 𝑚𝑜𝑑𝑒𝑙 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 𝑇𝑖𝑚𝑒 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠

𝐵𝑜𝑢𝑛𝑑𝑠 𝑜𝑛 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑡𝑖𝑚𝑒 𝑆𝑐ℎ𝑒𝑑𝑢𝑙𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠 𝐷𝑖𝑠𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 𝑚𝑜𝑑𝑒

(2)

The optimization problem solved in the NMPC

at the control layer is defined by Eq. 3. The decision

variables here are the mass flows in the

network (𝑚𝑡,𝑎∀𝑡 ∈ 𝑇𝑖𝑚𝑒, 𝑎 ∈ 𝐴𝑟𝑐𝑠). At this level,

the future cleanings are known and set as parameters

(from the upper layer), i.e. the future operating

modes of each exchanger is known a-priori.

Constraints include the network connectivity

constraints, the heat exchanger model and the

fouling model for all units, and the operational

constraints. Some common operational constraints



4

in this layer are: bounds on the split fraction of

parallel branches, pressure drop limits, and furnace

duty limits. This problem has significantly fewer

constraints than the optimal scheduling problem, Eq.

2, and no discrete variables, as the cleanings in this

layer are set. This is a dynamic optimization

problem, that after discretization of the differential

equations become a large-scale NLP with many

constraints and nonlinear terms, defined over short

prediction horizons (~days). In this case time is

represented using a discrete approach.

min𝑚

𝐽𝐶 = 𝐸𝑞. 3

𝑠. 𝑡. 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝐻𝑒𝑎𝑡 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒𝑟𝑠 𝑚𝑜𝑑𝑒𝑙 𝐹𝑜𝑢𝑙𝑖𝑛𝑔 𝑚𝑜𝑑𝑒𝑙 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑡𝑛𝑠

(3)

The previous two optimization problems are

parametric problems, i.e. their solution depends on a

set of parameters 𝑃𝑆 and 𝑃𝐶 for the scheduling and

control problem, respectively. Set 𝑃𝑆 includes the

initial model conditions (initial fouling resistance of

each exchanger), the fouling model parameters, and

the prediction horizon. Set 𝑃𝐶 includes the same

elements plus the definition of all cleanings within

the prediction horizon. The values of parameters in

these two sets are updated online according to the

plant measurements at a specific time, and become

inputs to the corresponding optimization problem.

However, the fouling model parameters are not

updated directly as they are obtained by solving a

parameter estimation problem with past

measurements.

The fouling model parameters for each

exchanger of the network are calculated by solving

the network-wide parameter estimation problem

defined by Eq. 4. This problem corresponds to the

MHE problem, and it is defined for each of the

layers, so there is a MHE for the scheduling layer,

and a MHE for the control layer. Here the objective

function is the minimization of the quadratic error

(difference between model and plant data) for the

primary measurements in the network. These

include tube side streams outlet temperature, shell

side streams outlet temperature, and tube side outlet

pressure, each weighted depending on their relative

importance. The problem is subject to the same

constraints as the previous optimization problems in

their respective layer. The solution of this single

problem defines the fouling parameters for each

exchanger in the network, which may vary from one

unit to the other. To avoid correlation-related

numerical difficulties, the activation energy is

assumed fixed and only the deposition constant (𝛼), the removal constant (𝛾), and the deposit roughness

(𝜖) are considered as degrees of freedom (fitting

parameters). The solution of this problem also

provides an estimation of the initial conditions

(initial fouling resistance) at the beginning of the

prediction horizon, for the scheduling, Eq. 2, and

control, Eq. 3, optimisations. The parameter

estimation problem includes the number (N) of data

points collected in the past estimation horizon

(PEH). The PEH for the control layer is shorter than

that of the scheduling layer, but both should be

proportional to the future prediction horizon (FPH)

of the corresponding problem. This ensures that

enough data is used to fit a model, so as to provide

accurate predictions over the future horizon.

min𝛼𝑖,𝛾𝑖,𝜖𝑖∀𝑖∈𝐻𝐸𝑋

𝐽 = 𝑤𝑇𝑡(𝑇𝑡,𝑖,𝑘 − �̂�𝑡,𝑖,𝑘)2

𝑖∈𝐻𝐸𝑋𝑘∈𝑁

+ 𝑤𝑇𝑠(𝑇𝑠,𝑖,𝑘 − �̂�𝑠,𝑖,𝑘)2

+ 𝑤𝑃(𝑃𝑡,𝑖,𝑘 − �̂�𝑡,𝑖,𝑘)2

𝑠. 𝑡. 𝑁𝑒𝑡𝑤𝑜𝑟𝑘 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝐻𝑒𝑎𝑡 𝑒𝑥𝑐ℎ𝑎𝑛𝑔𝑒𝑟𝑠 𝑚𝑜𝑑𝑒𝑙 𝐹𝑜𝑢𝑙𝑖𝑛𝑔 𝑚𝑜𝑑𝑒𝑙

(4)

ONLINE FOULING MITIGATION

The three problems formulated in the previous

section are used together in an on-line fouling

mitigation strategy that, by updating the models

using the past measurements from the process, is

able to handle disturbances and operational changes.

The diagram in Fig 3 summarizes data flows and

major functional activities in the overall scheme. It

is composed by the equivalent of two control loops:

an inner, fast one that defines the flow rate

distribution in the network every time new

information is available, and an outer, slow one that

defines the cleaning schedule over a future horizon.

The two loops address different time scales of the

process and allows responding efficiently to

unknown process changes and disturbances. The

inner loop responds faster and uses a shorter PEH

and FPH than the outer loop. Each loop corresponds

to one layer of decisions and solves recurrently a

NMPC and a MHE problem. Also, the PEH for the

MHE, and the FPH for the NMPC are different for

each loop. The MHE of the inner loop (control)

looks at data over just the past few days (short PEH)

to capture the current variability of the process, so

that the corresponding NMPC can quickly define the

flow distribution accordingly. On the other hand, the

MHE of the outer loop (scheduling) has a much

larger PEH and FPH, usually of the order of months,

because of the slow fouling rates of most units. Also,

a large FPH allows including future cleaning

actions, their interactions and trade-offs. As noted,

the control decision variables are also considered at

the outer loop as that NMPC problem can predict

optimal flow rates simultaneously with the optimal

cleaning scheduling, which has been demonstrated

to have a strong synergetic effect on fouling

mitigation [18]. However, these updates are much

slower than the variability of the process, and



5

therefore flow distribution variables are handled in

the inner loop. Only the optimal cleaning schedule

for the first prediction period, calculated by the outer

loop NMPC, is sent to the plant and inner loop

NMPC controller for implementation.

Fig 3. Control and scheduling diagram representing the receding horizon approach for online fouling mitigation.

The process is considered to be subject to

disturbances such as changes in the inlet flow rates,

inlet streams temperature, type of crude processed,

etc. Some of these disturbances can be measured,

and are included directly in the inner loop controller.

For the outer loop it is necessary to forecast

disturbances over long periods (~months). If

forecast functions are available, they can be used

directly in the NMPC algorithm for this layer.

Alternatively, a suitable disturbance model may be

used, for example a disturbance can be fixed at its

average or worst-case value over a past horizon, or

expected value over a future window, in the

optimization of future cleaning schedules. However,

known future events are readily incorporated.

In summary, this scheme updates the model

online, and defines cleaning schedules and the flow

distribution profiles over the entire network using a

receding horizon approach. For a successful

implementation of this methodology, the

optimization problems must solved in a much

shorter time than the sampling times, so no delay is

introduced.

All algorithms are implemented in Python,

using Pyomo 5.2 [19] as a modelling environment to

formulate and solve the optimization problems. For

more information about the formulation of the

optimization problems, and specially the solution

strategy of the integrated optimal scheduling and

control problem using complementarity constraints

the reader is referred to [7], [18].

CASE STUDY, RESULTS AND DISCUSSION

This online fouling mitigation methodology is

demonstrated for the case study depicted in Fig 1, a

heat exchanger network with four exchangers and

three parallel branches. Although small, it captures

the important features of industrial size ones:

multiple hot streams with different properties, flow

splits on the crude side, interaction between the

exchangers through the hot streams, a furnace to

supply the additional energy, and different design

specifications for the units. Only mechanical

cleanings and flow split control are considered. It is

assumed that the flow split can be freely

manipulated within a lower bound of 10% and an

upper bound of 50% to each branch. The case where

the flow split is defined by the pressure drop in the

branches can be included in the model and the

implementation. However, this reduces the degrees

of freedom associated with the flow distribution of

the network and is not considered here. Other

operational constraints are bounds on pressure drop

(hydraulic limit), maximum furnace duty, and the

number of units that can be simultaneously cleaned

(maximum 2). It is assumed that all the

measurements from the plant are available. Here, the

“real” plant is represented using the same prediction

model but with different parameters for the fouling

model. Also, these parameters can be changed

dynamically to mimic the variability of a real

operation.

Two scenarios are analyzed. Scenario 1 has

constant inputs and step changes in the inlet flow

rate of crude, with model-plant mismatch. Scenario

2 is more realistic, with dynamic inputs (variable

inlet flow rate and streams temperature) and again

model-plant mismatch. Both scenarios start with

clean conditions. Key operating conditions and

algorithm settings are summarized in Table 1.

Plant (‘real HEN’)

MHE – control layer

PEH ~ days

MHE – scheduling layer

PEH ~ months

NMPC – control layer

FPH ~ daysNMPC – scheduling layer

FPH ~ months

Forecast model for long

time disturbances

Disturbances

𝑚𝑎

𝑇𝑎𝑃𝑎

∀𝑎 ∈ 𝐴𝑟𝑐𝑠

𝑚𝑠

𝑇𝑠 ∀𝑠 ∈ 𝑆𝑜𝑢𝑟𝑐𝑒𝑠

𝐶𝑟𝑢𝑑𝑒 𝑡𝑦𝑝𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑖𝑒𝑠

𝛼𝑖

𝛾𝑖𝜖𝑖

𝑅𝑓 ,𝑖𝑜

∀𝑖 ∈ 𝐻𝐸𝑋

𝛼𝑖

𝛾𝑖𝜖𝑖

𝑅𝑓 ,𝑖𝑜

∀𝑖 ∈ 𝐻𝐸𝑋

𝑚𝑎∗∀𝑎 ∈ 𝐴𝑟𝑐𝑠

𝑦𝑡 ,𝑖∗

𝑡𝑠,𝑡 ,𝑖∗ ∀𝑡 ∈ 𝑇𝑖𝑚𝑒, 𝑖 ∈ 𝐻𝐸𝑋

𝑚𝑡 ,𝑎∗ ∀𝑡 ∈ 𝑇𝑖𝑚𝑒,𝑎 ∈ 𝐴𝑟𝑐𝑠

𝑚𝑠

𝑇𝑠 ∀𝑠 ∈ 𝑆𝑜𝑢𝑟𝑐𝑒𝑠

Control and scheduling loop – Slow sampling (~ 30 days)

Control loop – Fast sampling (~ 1 day)



6

Table 1. Key settings for the case study

Frequency of plant measurements 1 day

Duration of a cleaning 10 days

Cost of cleanings $30,000

Price of energy $ 27 / MW-h

Coil outlet temperature 350°C

Control layer

Future prediction horizon (FPH) 30 days

Past estimation horizon (PEH) 30 days

Sampling time 1 day

Scheduling and control layer

Future prediction horizon (FPH) 180 days

Past estimation horizon (PEH) 90 days

Sampling time 30 days

Scenario 1. Step disturbances in flow rate, and

model-plant mismatch.

The closed-loop performance of the network is

analyzed over 600 days of operation. The crude inlet

flow rate is set to 90 kg/s for the first 200 days,

decreased to 70kg/s for 200 days, and finally

increased to 110 kg/s for another 200 days. To test

for model mismatch, the deposition constant of the

fouling model (𝛼) of the plant is 10% larger than

that of the control model at the start of the operation.

Fig 4. Cleaning scheduling for case study - scenario

1. Step changes in the crude inlet flow rate

Fig 4 shows the cleaning schedule as

implemented, at the end of the 600 day period.

During execution, the number and assignment of

cleanings changes significantly when the crude flow

rate changes. Crude throughput changes the trade-

off between the cost of cleaning and benefit of future

energy recovery. High inlet crude rates favor more

frequent cleanings, while for low throughput

operations, cleanings are not favorable as their

higher cost is not offset by the future benefits from

higher energy recovery. Nonetheless there are more

cleanings during the period at a crude flow rate of

70 kg/s than at crude flow of 90 kg/s. This is because

after 200 days of operation there is already a built-

up deposit, and cleanings become advantageous.

The overall operating cost of the network over 600

days is $ 16.81 Millions, distributed as follows:

32.26 % for the initial 200 days operating at 90 kg/s,

25.86 % for 200 days at 70 kg/s, and 41.88 % for the

final 200 days at 110 kg/s.

Scenario 2. High variability input data.

This scenario mimics more closely the

operation of a heat exchanger network where the

type of crude processed and operating conditions

change frequently. Inlet flow rates and inlet streams

temperature are assumed to change daily. In the

prediction step, their variability is modelled as

normal or uniform distributions around the nominal

operating point. To simulate the effect of changing

the crude being processed, it is assumed that

different crudes have different fouling propensities.

These are represented as random changes in the

deposition constants of the fouling model in the

“real” plant simulation.

Fig 5. Cleaning scheduling for case study - scenario

2. Dynamic inputs

This is a much more challenging scenario as the

input dynamics make it more difficult to achieve an

accurate prediction of the network future

performance. However, the feedback loop, the data

sampling, and the online estimation of model

parameters handle this effectively. Fig 5 shows the

cleaning schedule achieved after 370 days of

operation. Despite the variability in the process, the

cleanings are regular, and the algorithms are able to

predict simultaneous or consecutive cleanings that

are advantageous to mitigate fouling. For example,

at time 100 days HEX1 and HEX2A are cleaned

simultaneously, and at time 300 those two

exchangers are cleaned sequentially starting with

HEX2A. Other methods that define the cleaning

schedule offline tend to agglomerate all the

cleanings towards the middle of the operation, with

no cleanings at the beginning nor at the end. This

online implementation reflects the current state of

the network, assumes a continuous operation and

distributes the cleanings better along the operating

time.

Regarding the optimal flow distribution in the

network, Fig 6 shows the split fraction between the

three parallel branches. At the beginning of the



7

operation, when there is no fouling, the branches are

balanced and the optimal split fraction is 0.33 to

each one. This changes with time as material starts

to deposit and the operating conditions change. The

algorithm interacts with the predicted cleanings by

modifying the split fraction, so that the minimum

flow is sent to a branch of an exchanger being

cleaned, and the rest is distributed optimally to

maximize energy recovery (or by passed in some

cases to meet the flow constraints). Operating

constraints such as the bounds on the split ratio are

satisfied at all times, in spite of the cleanings and

disturbances in the inlet streams.

Fig 6. Optimal flow split to parallel branches for

the case study - scenario 2

To illustrate the benefits of the proposed

approach, scenario 2 is simulated with the same

inputs variability and disturbances, but without any

fouling mitigation action. The split ratios are fixed

at 33.33% for each branch, and there are no

cleanings. Fig 7 shows a comparison between the no

mitigation case (NM) and the simultaneous online

optimal cleaning scheduling and control (O-SC) for

the CIT This figure clearly shows that the dynamic

inputs and changes in the crude oil introduce large

variability in the process and have a strong effect on

the network performance. In spite of disturbances, in

the NM case the CIT has a clear underlying

decreasing trend. By the end of the year the overall

decay in performance due to fouling results in a loss

of about 40°C in the CIT. On the other hand, the

online optimal solution maintains the CIT at

significantly higher levels for most of the operation.

Until the first cleaning (of HEX2B starting on day

61) the CIT for the two approaches is very similar.

Afterwards, the optimal cleanings and flow

distribution reduce significantly the performance

decay caused by fouling, with a loss of about 15°C

in the CIT by the end of the year. During the

cleaning periods, the CIT reaches lower levels

because of the lack of units in the network, but this

is compensated in the future by the restoration of the

heat transfer capabilities of the unit. In economic

terms, the total cost due to fouling for the NM

operating mode is $ 11.20 Millions. For the optimal

O-CS operation it is reduced to $ 10.83 Millions

(including furnace fuel saving of $ 0.8 Millions

against a cleaning cost of $0.39 Millions, a 100%

return on the cleaning investment). As operations

are assumed to continue after 365 days, the

economic benefits of the cleanings close to the final

time of the analysis (5 cleanings carried out after day

300) are not yet been accounted for.

CONCLUSION

A method and implementation for online

fouling mitigation in heat exchanger networks has

been presented, and a case study for a representative

refining application has demonstrated its

applicability and benefits. The method calculates the

optimal flow distribution in the network and the

optimal cleaning schedule simultaneously. The

optimal decision policies are updated online by

solving optimization problems at appropriate

frequencies. The prediction models used in the

optimizations are updated frequently, based on past

plant measurements.

The main conclusions of this study are:

1. the online operations optimization is very

effective in countering process variability.

2. the periodic update of prediction models in

a closed loop scheme ensures their validity.

3. the optimization formulation is very

powerful and flexible. It allows for the

simultaneous optimization of flow

distribution and cleaning schedules, with a

variety of constraints.

4. The proposed scheme reduces the total

operating cost of fouling significantly.

Fig 7. CIT for case study – scenario 2. Comparison of the no mitigation case (NM) and the optimal online

cleaning scheduling and control case (O-CS).



8

NOMENCLATURE

𝐾𝑒𝑟𝑜 Kerosene

𝑉𝑅 Vacuum residue

𝐿𝐺𝑂 Light gas oil

𝑁𝐿𝑃 Nonlinear program

𝑀𝐼𝑁𝐿𝑃 Mixed integer nonlinear program

𝐶𝐼𝑇 Coil inlet temperature

𝐻𝐸𝑋 Set representing all exchangers in a network

𝑚 Mass flow rate, kg/s

𝑚𝐶𝑂2Ratio of carbon emissions per energy, kg /W

𝑃 Unitary price

𝑃𝑡 Tube side outlet pressure, bar

𝑄𝑓 Furnace duty, MW

𝑡 Time, days

𝑢 Manipulated variables

𝑤 Relative weights in objective function

𝑥 General representation of the plant states

𝑦 Binary variable for cleanings definition

𝛼 Fouling deposition constant, m2K/W day

𝛾 Fouling removal constant, m4K/W N day

𝛿 Deposit thickness, mm

𝜆𝑑 Deposit thermal conductivity, W/m K

Subscript

0 Initial condition

𝐶 Optimal control problem

𝑓 Final condition

𝑠 Starting time

𝑆 Optimal cleaning scheduling problem

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